172 ANALYTICAL MECHANICS 13. A vertical shaft carries a load of 10 tons. Find the work lost per revolution if the shaft is 8 inches in diameter and has a flat-end bearing; the coefficient of friction being 0.01. 14. Derive an expression for the work done in giving a collar-bearing pivot an angular displacement. 15. A vertical shaft carries a load of 5 tons. Find the work lost per revolution if the shaft is supported by a collar-bearing pivot which has an inner diameter of 6 inches and an outer diameter of 8 inches. The coefficient of friction is 0.1. HOOKE'S LAW. 133. Stress. — When a body is acted upon by external forces which, tend to change its shape and thus give rise to forces between its contiguous elements, the body is said to be under stress. The measure of stress is force per unit area: where S denotes the stress, F the force, and A the area on which the latter acts. 134. Pressure, Tension, and Shear. — Stresses which occur in bodies are often of a complex nature, but they may be resolved into three component stresses of simple character. These are called pressure, tension, and shear. Pressure tends to compress, tension to extend, and shear to distort bodies. Shearing stress is the result of a compressive stress combined with a tensile stress at right angles. A special case of shear, which comes into play within a shaft when the latter is twisted, is called torsion. 135. Strain. — Strain is the deformation produced by stress. The measure of strain is the percentage deformation. For instance, if the deformation consists of a change in length the strain equals the ratio of the increase in length, to the original length : 7 «-r (v) where s denotes the stress, I the increase in length, and L the original length.